1. Images

2. Pinhole Lens Photons from a source can be focused by a small aperture. –Aperture radius a –Magnification m –Image is inverted Image is blurry –Blurred size w –Sharper with smaller aperture –Diffraction limited h’ vu object planeimage plane h

3. Pinhole Radiance The power from a pinhole is restricted. –Radiance L –Area dS –Power P –Irradiance E This is related to the solid angle subtended by the pinhole at the image.

4. Point Spread The general relation between radiance and irradiance includes blurring and aberration. –Point spread function h –Equivalent to image from a point source The function can be described analytically for an ideal lens.

5. Convolution The irradiance is a two- dimensional convolution. There is a theorem that relates the transforms of a convolution. –Cosine transform C(k) –Sine transform S(k) This is the optical transfer function (OTF).

6. Transfer Functions The amplitude and phase of the OTF can be expressed. –Amplitude: modulation transfer function (MTF) –Phase: phase transfer function (PTF)

7. Modulation Radiance from an incoherent source has dc and ac components. The image can be described in terms of the transfer function. The modulation compares the minimum and maximum radiance.

8. Image Quality Optical image quality depends on three properties. –Resolution: minimum size that can be identified. –Contrast: difference in light from adjacent areas –Noise: Fluctuations due to statistics or background.

9. Contrast Contrast look at the change in light compared to the average. –Analogous to modulation Exposure contrast compares light coming in. –Exposure X is energy fluence of beam. Brightness contrast compares light going out. –Transmittance T

10. Poisson Noise A uniform source will produce a random count of photons in an area. –Fraction f will interact –Quantum efficiency The noise contrast is based on the standard deviation of the noise. –Noise exposure contrast –Noise brightness contrast For a fluence  in an area S The noise contrast is

11. Signal to Noise Signal is also a fluctuation compared to the background. Signal-to-noise is compares the signal contrast to the noise contrast. A signal is detectable when S/N > k, a detection constant. –Perception k ~ 2 to 5

12. X-Ray Imaging Typical Problem A monoenergetic x-ray beam, fluence  0, passes through an object of thickness L, and attenuation coefficient . One region has reduced attenuation  for length x. What is the signal-to-noise ratio? Answer The exiting fluence   in each region is Assume  x is small